12. Predicate Logic Structures. The Lecture
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1 12. Predicate Logic Structures The Lecture
2 What is predicate logic?
3 What is predicate logic? Predicate logic deals with properties of elements and relations between elements of a domain.
4 What is predicate logic? Predicate logic deals with properties of elements and relations between elements of a domain. We can talk about universal properties and existence of solutions of equations.
5 What is predicate logic? Predicate logic deals with properties of elements and relations between elements of a domain. We can talk about universal properties and existence of solutions of equations. A basic concept is that of a structure, also called a model.
6 Unary structure
7 Unary structure A unary structure M consists of a domain M and a number of subsets of it, called predicates. The predicates are denoted A 0,A 1,...
8 Unary structure A unary structure M consists of a domain M and a number of subsets of it, called predicates. The predicates are denoted A 0,A 1,... M A 0 One predicate divides the domain into up to two parts
9 Unary structure A unary structure M consists of a domain M and a number of subsets of it, called predicates. The predicates are denoted A 0,A 1,... M A 0 One predicate divides the domain into up to two parts M Two predicates divide the domain into up to four parts A 0 A 1
10 Examples
11 Examples M= a set of people A 0 = the set of women in M M A 0 Men Women
12 Examples M Men M= a set of people A 0 = the set of women in M A 0 Women M= a set of people A 0 = the set of country music lovers in M A 1 = the set of jazz fans in M M A 0 A 1 Likes country music Likes jazz
13 Unary structure with three predicates divides the domain into up to 8 parts. M A 0 A 1 A 2
14 Example Likes classical music Likes classical music and jazz Likes classical and country music Likes jazz Likes country music Likes country music and jazz
15 Tile models
16 Tile models A tile model consists of colored tiles arranged in a row as the five tiles below:
17 Tile models A tile model consists of colored tiles arranged in a row as the five tiles below: The relevant properties of the tiles are: Color. Position: which is left or right of which.
18 Examples of tile models
19 Examples of tile models
20 Examples of tile models
21 Examples of tile models
22 Examples of tile models
23 Examples of tile models
24 A mathematical definition of tile models
25 A mathematical definition of tile models A tile model T consists of
26 A mathematical definition of tile models A tile model T consists of a finite set T of tiles
27 A mathematical definition of tile models A tile model T consists of a finite set T of tiles For each tile x exactly one of the predicates B T (x) x is blue, R T (x) x is red, Y T (x) x is yellow holds.
28 A mathematical definition of tile models A tile model T consists of a finite set T of tiles For each tile x exactly one of the predicates B T (x) x is blue, R T (x) x is red, Y T (x) x is yellow holds. There is a linear order < T defined on T. If x < T y, we say x is left of y and y is right of x, and write x< T y.
29 A mathematical definition of tile models A tile model T consists of a finite set T of tiles For each tile x exactly one of the predicates B T (x) x is blue, R T (x) x is red, Y T (x) x is yellow holds. There is a linear order < T defined on T. If x < T y, we say x is left of y and y is right of x, and write x< T y. A linear order on a finite set is a specification of the order of the elements: which is the first, which comes next, etc.
30 Graphs
31 Graphs A graph consists of vertices and edges between the vertices as in:
32 Graphs A graph consists of vertices and edges between the vertices as in:
33 Graphs A graph consists of vertices and edges between the vertices as in: In this picture vertices are blue, edges are red.
34 Graphs A graph consists of vertices and edges between the vertices as in: In this picture vertices are blue, edges are red. Graphs are common in applications.
35 Graphs A graph consists of vertices and edges between the vertices as in: In this picture vertices are blue, edges are red. Graphs are common in applications. Vertices connected by an edge are neighbors.
36 More graphs
37 More graphs
38 More graphs
39 More graphs
40 More graphs
41 More graphs
42 More graphs
43 A mathematical definition of graphs
44 A mathematical definition of graphs A graph G consists of
45 A mathematical definition of graphs A graph G consists of a domain G, called the set of vertices, and
46 A mathematical definition of graphs A graph G consists of a domain G, called the set of vertices, and a binary predicate xey (more exactly xe G y) for the edge relation. Then x is called a neighbor of y and vice versa.
47 A mathematical definition of graphs A graph G consists of a domain G, called the set of vertices, and a binary predicate xey (more exactly xe G y) for the edge relation. Then x is called a neighbor of y and vice versa. No vertex is a neighbor of itself. (Antireflexivity)
48 A mathematical definition of graphs A graph G consists of a domain G, called the set of vertices, and a binary predicate xey (more exactly xe G y) for the edge relation. Then x is called a neighbor of y and vice versa. No vertex is a neighbor of itself. (Antireflexivity) If xey then yex. (Symmetry)
49 The integers
50 The integers The natural numbers are the nonnegative integers 0,1,2,...
51 The integers The natural numbers are the nonnegative integers 0,1,2,... They have a natural order < in which 0 is the smallest element and for every element x there is a bigger one, namely x+1.
52 Other structures (some with functions) Directed graph Equivalence relation Group Field Boolean algebra Lattice Linear order Partial order Tree
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